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𝘽𝙧𝙤𝙩𝙝𝙚𝙧, 𝙄 𝙬𝙖𝙣𝙩 𝙩𝙤 𝙡𝙚𝙖𝙧𝙣 𝘿𝙄𝙁𝙁𝙀𝙍𝙀𝙉𝙏𝙄𝘼𝙇 𝙂𝙀𝙊𝙈𝙀𝙏𝙍𝙔...... 𝙖𝙣𝙙 due to absence of right guider, I am unable to learn it...... I am from India🇮🇳....... Where are you from?
@@Mad_mathematician224 bro what are you begging for, you have access to the internet. Courses: google -> differential geometry -> MIT OpenCourseWare Textbooks: google + pdf -> download links -> books Simple as
Dude just wanted to thank you soo much for your videos, they've helped me gain a profound interest in maths at higher levels even though I'm still in school lol. Also, I'd love yo here your thoughts about topics like other Millienuem(spelling wrong ik) problems or even the Langlands Project. Thanks again for everything!
@@just.a.random.ava.-_- I'm very glad to hear that! There's definitely more number theory / Langlands videos + Millennium problem videos coming up soon, so keep your eyes peeled :)
The reason exp(1/Z) contains an essential singularity is, if you expand the function as a Taylor series, you will get infinitely many powers of (1/Z). In essence, the singularity can't be removed by multiplying by Z. Therefore, it is "essential"
Another fact about essential singularities: A function with an essential singularity takes all complex values (or all complex values except one value) infinitely many times in every open neighborhood of the essential singularity (Picard's Great Theorem)
Or more directly, as z goes to 0 from the positive real direction, 1/exp(1/z) goes to 0, but as z goes to 0 from the negative real direction, 1/exp(1/z) goes to infinity. So 1/exp(1/z) can't be continuously extended to 0 even in the real line, let alone the complex plane.
@@EebstertheGreat that's true of functions with poles like 1/z^n at 0 too. The point is that the singularity exp(1/z) has at 0 is not that simple, in the sense that it can't be removed by multiplying it with some z^n - hence the name essential.
One of my favorite things in complex analysis was just seeing that elliptical curve come out of nowhere with the Weierstrass p-function, I felt like I was seeing a fraction of what Wiles saw every day while proving the modularity theorem enough to prove Fermat’s last conjecture.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
@hybmnzz2658 I don't think I'd say Falting's theorem says anything much about interesting group laws, but maybe I see why you ask this question. Here are some simpler points about group laws you can say. The degree-genus formula tells you that for degree >4 the genus of the curve in P^2 is atleast 3 (which in particular is > 1). If the curve is additionally defined over the rationals, the K-points C(K) are finite. So if C were a group scheme defined over Q, then C(K) would have to be a finite subgroup. There is no obvious reason this gives a contradiction though, but actually theres a much easier reason why any 1-dimensional group scheme over Q is actually genus 1: The group scheme structure allows you to give a trivialisation of the tangent bundle (as the translation action of C on itself is transitive on Q-points). The only smooth connected curve over Q with trivial tangent bundle is genus 1, since the degree of the tangent bundle is 2g - 2.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
Aleph 0 is back with yet another banger ! Nah but seriously as a grad student in applied analysis/probability/statistics and little knowledge of pure maths, i enjoy these videos so much as they give me a glimpse of the beauty of what's on "the other side". Please keep them coming !
This is an extremely good motivation for the elliptic curve equation(s) that I see everwhere, and a very nice explanation why complex tori are elliptic curves (and not just the other way around)! I'm a bit baffled by your way to write a zeta though...
This is one of my favorite results in mathematics. I used this categorical equivalence (along with the equivalence to algebraic function fields) in my master's thesis on prime Galois coverings of the Riemann sphere back in 2007.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
my math is such a rust bucket. i need to dust off a bunch of old books, but then recapitulate several semesters just to be sure i had enough of the definitions fixed in my head
Thanks for the video, very well explained! On this topic, I found the book by Serge Lang on elliptic functions very helpful, but also Gunning's lectures on Riemann surfaces for every thing beyond genus 1
Great amazing content, I admire the effort that went into making this!!! I would add a short section about the inversion 1/z (with animation) to explain the essential singularity at infinity.
Exponentials are dual to logarithms. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
I don’t know if it’s important, but in the complex torus example the interval is first written as closed [0,2pi] and later in the example it is written as open [0,2pi).
Continuous (classical) is dual to discrete (quantum). Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
Something that gets lost in Riemann's immense contribution to humanity was the shockingly forward thinking idea he introduced that the microscopic spacetime may be nothing like the 3 + 1 we know so well, over a hundred years before Dirac postulated the same thing which is basically where theoretical physics is now.
Space is dual to time -- Einstein. Time dilation is dual to length contraction -- Einstein, special relativity. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
I would love a video on GAGA theorem (Serre), which is really a continuation on the topic in this. It is remarkable how Riemann's work in the late 1800's is the foundation for modern algebraic geometry.
I think it's because the curve on the right is actually in C2. Like how in the previous example t from the interval which is in R gets mapped to the circle in R2 by associating the points (x,y) on the circle with t on the interval via the trig functions ie x= cos t and y = sin t. ... In the same way, each complex pair (X, Y) on the "curve" described on the right is associated with a complex number z in the square via the functions X = P(z) and Y = P'(z).
The fact that there _is_ a connection between complex numbers and geometry isn't shocking at all (a very obvious connection is spinny), but I can say that I wasn't aware of this particular connection.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
At 3:20, doesn’t the zeta function have an essential singularity at infinity? Edit: Oh, you meant that the functions on the left are *not* meromorphic at infinity
Points are dual to lines -- the principle of duality in geometry. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
I recently learned Riemann surfaces are used in string theory which I find really cool. I also am 90% sure they come up in the 2-spinor formalism of GR but it’s never clicked for me
Action is dual to reaction -- Lagrangians are dual, forces are dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
Why not just define f(infinity) to be the limit as z approaches the point at infinity of f(z) where we can take |z| approaching infinity in the real case and consider all possible paths of z that do this. Why would these two limits not be the same when they exist?
lets say z=x+iy... Where relations like f(z) = e^(z^2/(z^2-1)) unitstep(1-z^2) fall inside complex analysis? If y=0 then f(z) it is a smooth bump function, which are not analytic so at least in the real line f(z) cannot be represented as a power series, which rule it out of conventional complex calculus (this is why I call it a relation instead of a function). There is a branch of mathematics that study this kind of complex-valued objects?
This kind of thing falls more into the realms of real analysis in multiple dimensions. Functions that aren't analytic aren't complex-differentiable. You may be able to define such functions using complex numbers, but the algebraic structure of the complex numbers isn't really relevant for understanding these functions. It's more useful to rewrite these functions from R^2 to R^2 and study them using tools from real analysis (which includes standard multivariable calculus).
@@chobes1827 and how it is done? do you know how this kind of analysis is named?... At least for me is not obvious how you will make happen in R^2 all the oscillating effects that rises from Euler identity e^(it)=cos(t)+i sin(t) without it, my example f(z) it is just a 2D smooth bump function, but I think it is not his complex behaviour since in their exponent the z^2 term will left some terms dependent in the imaginary unit "i", leading to oscillating behaviour
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
@whatitmeans You basically just write that g((x,y)) = (Re(f(x+iy), Im(f(x+iy)) and then study g as a map from R^2 to R^2. Points in the complex plane are still just pairs of two real numbers, and you can always identify them as such. If you study some complex analysis, you'll learn how this works because you need to think about complex functions this way in order to derive and use the Cauchy-Riemann equations. All of the oscillating behavior ends up being expressed with rotation matrices, and it's all completely doable despite the expressions being a bit messier. For example, if r is |z| and theta is the angle formed between z and the positive real axis, then e^iz becomes e^r * rotation by theta as a map from R^2 to R^2.
Very interesting! I never got into complex analysis in uni. Can I suggest you just stick with Canadian ‘zed’? I think American viewers will understand :)
My first time watching produced a question that, isn’t then the conclusion made in the end (analytical object can be bridged to algebraic perspective) requires the step glossed over in “projective complex space” CP2 thing, that dealt with how 0 is mapped? I might watch the second time but maybe someone was impressed with the same question as me in the comment section.
"Complex analysis is algebraic geometry in disguise." Given that analytic functions can be described as glorified polynomials, that kind of gave a hint. (Am I seeing that correctly?)
You're exactly right about that. The big idea is really that if you look at analytic and meromorphic functions ("glorified" polynomials and rational functions respectively) that satisfy very natural conditions, they turn out to be polynomial or rational.
Space is dual to time -- Einstein. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
@@hyperduality2838, it's you!!! You and I have had many great conversations across TH-cam! I actually have a spreadsheet where I have kept a running list of your awesome examples of duality. I have found them incredibly profound ... and helpful.
This math might be above my level, but it makes me hopeful. I was just wondering how to approach classifying ranges of relations as Rational or Irrational.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
So is this like the Langlands program, just for Complex Analysis and Algebraic Geometry, as opposed to Number Theory and Geometry? Just trying to get my head around these different branches of Mathematics of which I clearly know so little!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
I call them "spinny numbers", because they are the best tool for the job of making 2D objects go spinny. (Naturally there are also 3D spinny numbers, which are rather famous, or more accurately infamous.)
5:33 Why do we need to explicitly evoke f(1/z)? Will lim z->inf f(z) not work? Also, to check if a function is meromorphic at inf, is there no other way than to see this other than checking singularity of f(1/z)?
The notion of taking a limit as a value approaches infinity isn't well defined in the complex plane the same way it is for the real line. On the real line, there's only really one way we can make a variable approach infinity (by making the variable bigger and bigger). In the complex plane, variables can grow infinitely along an uncountably infinite amount of paths that move in different directions. We need to make a statement about what happens as z grows infinitely large in any of the possible directions. We're interested in what happens as |z| approaches infinity along any possible path. Working with lim |z| -> infinity is technically sufficient to formulate the definition of a function being continuous, holomorphic, or meromorphic at infinity, but it's tricky to reason about a variable growing larger across the entire plane. We use the fact that as |z| approaches infinity, |1/z| approaches 0 to make the behavior we're interested in easier to reason about. By looking at the behavior of f(1/z) when |z| is small, we can study the behavior of f(z) as |z| approaches infinity by reasoning about the behavior of a function on a small disk, which is much more manageable than thinking about f's behavior as z grows larger in any of the possible directions.
Exponentials are dual to logarithms. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
@@heeraksharma1224 Points are dual to lines -- the principle of duality in geometry. If Riemann geometry is dual then this means that singularities (points) are dual. Black holes = positive curvature singularities. White holes (the big bang) = negative curvature singularities. The definition of Gaussian negative curvature requires two dual points:- en.wikipedia.org/wiki/Gaussian_curvature The big bang is an infinite negative curvature singularity -- a Janus point/hole. Two faces = duality. The physicist Julian Barbour has written a book about Janus points/holes. Topological holes cannot be shrunk down to zero -- non null homotopic. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (a pringle) -- inflation. The big bang an explosion is repulsive by definition -- negative curvature. The point duality theorem is dual to the line duality theorem -- universal hyperbolic geometry. The bad news is that Einstein threw his negative curvature solutions in the proverbial waste paper bin of history!
Interesting choice to talk about the complex torus and the p function and y^2 = x^3-x and NOT mention the term Elliptic curve 😉 Guess you didn't want to overload the video with even more topics
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
Just checking but like. The arrow in that last image doesn’t go both ways, does it? Sure, every Riemann surface is an algebraic surface and that’s cool, but like. There are three-dimensional algebraic surfaces, but there are no three-dimensional Riemann surfaces, right? So there are some varieties that are not Riemann-able.
Is the final statement of this video false? Shouldn’t it be “SOME OF Complex Analysis SOME OF Algebraic Geometry”? Or do I need to watch the video again?
So every compact Riemann surface is an algebraic curve but is the other way true that every algebraic curve can be realized as a compact Riemann surface? If not these fields aren’t the same, just that these surfaces can be viewed equivalently in both but not all algebraic curves can be studied using complex analysis and not everything in complex analysis is a compact Riemann surface that can be studied in algebraic geometry. Therefore, I’m not sure I understand what the video is trying to conclude about them being the same and I’m just trying to understand that last point.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
Riemann geometry is dual. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
@@oscargr_ Points are dual to lines -- the principle of duality in geometry. If Riemann geometry is dual then this means that singularities (points) are dual. Black holes = positive curvature singularities. White holes (the big bang) = negative curvature singularities. The definition of Gaussian negative curvature requires two dual points:- en.wikipedia.org/wiki/Gaussian_curvature The big bang is an infinite negative curvature singularity -- a Janus point/hole. Two faces = duality. The physicist Julian Barbour has written a book about Janus points/holes. Topological holes cannot be shrunk down to zero -- non null homotopic. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (a pringle) -- inflation. The big bang an explosion is repulsive by definition -- negative curvature. The point duality theorem is dual to the line duality theorem -- universal hyperbolic geometry. The bad news is that Einstein threw his negative curvature solutions in the proverbial waste paper bin of history!
Exponentials are dual to logarithms. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Real is dual to imaginary -- complex numbers are dual. Injective is dual to surjective synthesizes bijective or isomorphism -- group theory. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Lie groups are dual to Lie algebras. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Riemann geometry or curvature is dual -- upper indices are dual to lower indices. Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry. Subgroups are dual to subfields -- the Galois correspondence. Elliptic curves are dual to modular forms. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory. "Always two there are" -- Yoda. Poles (eigenvalues) are dual to zeros -- optimized control theory. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Duality creates reality!
![Imaginary Numbers Are Real [Part 13: Riemann Surfaces]](http://i.ytimg.com/vi/4MmSZrAlqKc/mqdefault.jpg)
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12:05 Little spelling mistake, but Reimann is not going to mind.
I recommend TH-camr Richard Borcherds, who has multiple series about these.
𝘽𝙧𝙤𝙩𝙝𝙚𝙧, 𝙄 𝙬𝙖𝙣𝙩 𝙩𝙤 𝙡𝙚𝙖𝙧𝙣 𝘿𝙄𝙁𝙁𝙀𝙍𝙀𝙉𝙏𝙄𝘼𝙇 𝙂𝙀𝙊𝙈𝙀𝙏𝙍𝙔...... 𝙖𝙣𝙙 due to absence of right guider, I am unable to learn it...... I am from India🇮🇳....... Where are you from?
@@Mad_mathematician224 bro what are you begging for, you have access to the internet.
Courses: google -> differential geometry -> MIT OpenCourseWare
Textbooks: google + pdf -> download links -> books
Simple as
Dude just wanted to thank you soo much for your videos, they've helped me gain a profound interest in maths at higher levels even though I'm still in school lol. Also, I'd love yo here your thoughts about topics like other Millienuem(spelling wrong ik) problems or even the Langlands Project. Thanks again for everything!
@@just.a.random.ava.-_- I'm very glad to hear that! There's definitely more number theory / Langlands videos + Millennium problem videos coming up soon, so keep your eyes peeled :)
The reason exp(1/Z) contains an essential singularity is, if you expand the function as a Taylor series, you will get infinitely many powers of (1/Z). In essence, the singularity can't be removed by multiplying by Z. Therefore, it is "essential"
Another fact about essential singularities:
A function with an essential singularity takes all complex values (or all complex values except one value) infinitely many times in every open neighborhood of the essential singularity (Picard's Great Theorem)
Or more directly, as z goes to 0 from the positive real direction, 1/exp(1/z) goes to 0, but as z goes to 0 from the negative real direction, 1/exp(1/z) goes to infinity. So 1/exp(1/z) can't be continuously extended to 0 even in the real line, let alone the complex plane.
@@EebstertheGreat that's true of functions with poles like 1/z^n at 0 too. The point is that the singularity exp(1/z) has at 0 is not that simple, in the sense that it can't be removed by multiplying it with some z^n - hence the name essential.
Love this explanation! It's "essential" because you can't get rid of it by multiplying by Z. Brilliant.
This def includes removable and poles of any order, just number of terms that diverge, and 0 if removable
it’s the main argument of my thesis, I’m so happy to see a video about Riemann Surface ❤️
No better way to start a day than an aleph0 upload
What time zone are you in?
@@diaz6874 Australia, was 6am for me when this dropped
@@diaz6874 Australia, was 6 am for me when this dropped
@@diaz6874 glue 6 am to 2 pm. Geometry in algebraic disguise.
One of my favorite things in complex analysis was just seeing that elliptical curve come out of nowhere with the Weierstrass p-function, I felt like I was seeing a fraction of what Wiles saw every day while proving the modularity theorem enough to prove Fermat’s last conjecture.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
@hybmnzz2658 I don't think I'd say Falting's theorem says anything much about interesting group laws, but maybe I see why you ask this question. Here are some simpler points about group laws you can say. The degree-genus formula tells you that for degree >4 the genus of the curve in P^2 is atleast 3 (which in particular is > 1). If the curve is additionally defined over the rationals, the K-points C(K) are finite. So if C were a group scheme defined over Q, then C(K) would have to be a finite subgroup. There is no obvious reason this gives a contradiction though, but actually theres a much easier reason why any 1-dimensional group scheme over Q is actually genus 1: The group scheme structure allows you to give a trivialisation of the tangent bundle (as the translation action of C on itself is transitive on Q-points). The only smooth connected curve over Q with trivial tangent bundle is genus 1, since the degree of the tangent bundle is 2g - 2.
@@hyperduality2838actually the co in cosine is for cotermenal angle sine so sine (x+90)
When the world needs him he will come back
When the world needs someone, Surfshark brings him back
I needed this video today and he didn’t disappoint.
Amen
😂 @@phenixorbitall3917
😂@@phenixorbitall3917
Riemann’s existence theorem: “Bernhard Riemann exists.”
Do one on Donaldson theory!
Uh, I don’t know how to break this to you… but about Riemann existing…
Now can you explain Riemann’s mapping theorem.
@@billcook4768 Riemann’s mapping theorem:
Bernhard Riemann was a cartographer.
(This theorem is known to be false)
@@jakobr_😂😂😂
Finally, more Algebraic Geometry content
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
Aleph 0 is back with yet another banger ! Nah but seriously as a grad student in applied analysis/probability/statistics and little knowledge of pure maths, i enjoy these videos so much as they give me a glimpse of the beauty of what's on "the other side". Please keep them coming !
i don’t often comment on uploaded videos, but i feel this video is so good that i just wanted to say thank you, and keep up the good work.
I love how you give equal time to "zee" and "zed" 😅
This is an extremely good motivation for the elliptic curve equation(s) that I see everwhere, and a very nice explanation why complex tori are elliptic curves (and not just the other way around)!
I'm a bit baffled by your way to write a zeta though...
9:00 sick blotter design, bro :)
I've heard that blotter with Weierstrass elliptic function on it, kicks stronger
It'll have you seeing a point at infinity
This is one of my favorite results in mathematics. I used this categorical equivalence (along with the equivalence to algebraic function fields) in my master's thesis on prime Galois coverings of the Riemann sphere back in 2007.
Respect. The printed cut outs are beautiful.
Love the channel and the content, no pressure, but I have been eagerly awaiting the course that you talked about developing/releasing.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
Been waiting for a new video from you. Just checked a few days ago. And there it is. I'm already intrigued.
my math is such a rust bucket. i need to dust off a bunch of old books, but then recapitulate several semesters just to be sure i had enough of the definitions fixed in my head
Get some flashcards and set aside an hour a day. Start with something you love. You got it buddy ❤️
Check out 3Blue1Brown
Thanks for the video, very well explained!
On this topic, I found the book by Serge Lang on elliptic functions very helpful, but also Gunning's lectures on Riemann surfaces for every thing beyond genus 1
Great amazing content, I admire the effort that went into making this!!!
I would add a short section about the inversion 1/z (with animation) to explain the essential singularity at infinity.
Exponentials are dual to logarithms.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
I'm a simple person: first I like the new Aleph0 video, then I watch it (even hours later). Trust is everything!
@user-ky5dy5hl4d I ignore what you mean, but considering it’s Aleph0, he has all my trust for he is a brilliant mathematician.
Honey wake up, Aleph 0 just uploaded a new video
Great video! I would love to hear some more about this Weierstrass p function.
I don’t know if it’s important, but in the complex torus example the interval is first written as closed [0,2pi] and later in the example it is written as open [0,2pi).
Oh I was just watching your video on the continuum hypothesis! Nice to see you back!
Continuous (classical) is dual to discrete (quantum).
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
Absolute banger as always. I'm interested in making educational math content as well and I've been using you as inspiration for my pedagogy.
Any chance your name is Steven
@@brian.westersauce no his name is Brian.
@@primenumberbuster404 no his name is buster
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
Something that gets lost in Riemann's immense contribution to humanity was the shockingly forward thinking idea he introduced that the microscopic spacetime may be nothing like the 3 + 1 we know so well, over a hundred years before Dirac postulated the same thing which is basically where theoretical physics is now.
Space is dual to time -- Einstein.
Time dilation is dual to length contraction -- Einstein, special relativity.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
love your content. Please make a video about riemann hypotheses or more about the millenium problems. The biggest unsolved problems in math
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
Great to see more algebraic geometry!
"Sorry not now babe Aleph 0 just dropped"
I would love a video on GAGA theorem (Serre), which is really a continuation on the topic in this. It is remarkable how Riemann's work in the late 1800's is the foundation for modern algebraic geometry.
That’s a great suggestion. GAGA is definitely on the list for a future video!
10:00 Probably a really dumb question, but how does a square which is 2D become an algebraic curve which is 1D?
I think it's because the curve on the right is actually in C2. Like how in the previous example t from the interval which is in R gets mapped to the circle in R2 by associating the points (x,y) on the circle with t on the interval via the trig functions ie x= cos t and y = sin t.
... In the same way, each complex pair (X, Y) on the "curve" described on the right is associated with a complex number z in the square via the functions X = P(z) and Y = P'(z).
The curve is 1D over the complex numbers, which translates to 2D on the real numbers.
The fact that there _is_ a connection between complex numbers and geometry isn't shocking at all (a very obvious connection is spinny), but I can say that I wasn't aware of this particular connection.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
I would like to undersrand better if the etale space of the sheaf of holomorphic functions on a Riemann surface give another Riemann surface
At 3:20, doesn’t the zeta function have an essential singularity at infinity?
Edit: Oh, you meant that the functions on the left are *not* meromorphic at infinity
Points are dual to lines -- the principle of duality in geometry.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
I recently learned Riemann surfaces are used in string theory which I find really cool. I also am 90% sure they come up in the 2-spinor formalism of GR but it’s never clicked for me
Action is dual to reaction -- Lagrangians are dual, forces are dual.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
Beautiful animations but I would have liked more explanation of the basic concepts.
Nice, could you make something embracing all the symmetries of our beloved R3 smooth sphere?
Why not just define f(infinity) to be the limit as z approaches the point at infinity of f(z) where we can take |z| approaching infinity in the real case and consider all possible paths of z that do this. Why would these two limits not be the same when they exist?
lets say z=x+iy... Where relations like f(z) = e^(z^2/(z^2-1)) unitstep(1-z^2)
fall inside complex analysis?
If y=0 then f(z) it is a smooth bump function, which are not analytic so at least in the real line f(z) cannot be represented as a power series, which rule it out of conventional complex calculus (this is why I call it a relation instead of a function).
There is a branch of mathematics that study this kind of complex-valued objects?
This kind of thing falls more into the realms of real analysis in multiple dimensions. Functions that aren't analytic aren't complex-differentiable. You may be able to define such functions using complex numbers, but the algebraic structure of the complex numbers isn't really relevant for understanding these functions.
It's more useful to rewrite these functions from R^2 to R^2 and study them using tools from real analysis (which includes standard multivariable calculus).
@@chobes1827 and how it is done? do you know how this kind of analysis is named?... At least for me is not obvious how you will make happen in R^2 all the oscillating effects that rises from Euler identity e^(it)=cos(t)+i sin(t)
without it, my example f(z) it is just a 2D smooth bump function, but I think it is not his complex behaviour since in their exponent the z^2 term will left some terms dependent in the imaginary unit "i", leading to oscillating behaviour
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
@whatitmeans You basically just write that g((x,y)) = (Re(f(x+iy), Im(f(x+iy)) and then study g as a map from R^2 to R^2. Points in the complex plane are still just pairs of two real numbers, and you can always identify them as such.
If you study some complex analysis, you'll learn how this works because you need to think about complex functions this way in order to derive and use the Cauchy-Riemann equations.
All of the oscillating behavior ends up being expressed with rotation matrices, and it's all completely doable despite the expressions being a bit messier. For example, if r is |z| and theta is the angle formed between z and the positive real axis, then e^iz becomes e^r * rotation by theta as a map from R^2 to R^2.
Very interesting! I never got into complex analysis in uni. Can I suggest you just stick with Canadian ‘zed’? I think American viewers will understand :)
Thank you for saying zed :)
Kind of hope you could maintain the handwriting style. You in fact inspired me to do all-handwriting demos.
I wonder content of this type is also available love your content ❤.
at 5:00 the unit circle is labelled at points +/-1 and +/-I. wouldn't it make more sense if it was +/-1 and +/-i^2? thx.
Can you please add Thanks option to your videos.
Great video
Do you have any plans to make a video about p vs np?
My first time watching produced a question that, isn’t then the conclusion made in the end (analytical object can be bridged to algebraic perspective) requires the step glossed over in “projective complex space” CP2 thing, that dealt with how 0 is mapped? I might watch the second time but maybe someone was impressed with the same question as me in the comment section.
Mom! Mom! New Aleph0 video dropped🎉
A minor and not-so-important correction: the weirstrass p function and its derivative satisfy a polynomial equation with a constant term.
"Complex analysis is algebraic geometry in disguise." Given that analytic functions can be described as glorified polynomials, that kind of gave a hint. (Am I seeing that correctly?)
You're exactly right about that. The big idea is really that if you look at analytic and meromorphic functions ("glorified" polynomials and rational functions respectively) that satisfy very natural conditions, they turn out to be polynomial or rational.
@@chobes1827, thanks! Wow, what a fun video. It's always so satisfying to see new connections that are sitting _right there._
Space is dual to time -- Einstein.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
@@hyperduality2838, it's you!!! You and I have had many great conversations across TH-cam! I actually have a spreadsheet where I have kept a running list of your awesome examples of duality. I have found them incredibly profound ... and helpful.
Thank you!
Always glad to see you return
Great video!
neat to see Dhruv's name again in an unexpected place
12:15 it says Reimann but it should be Riemann. Other than that, thank you for the great video :D
I wonder how Riemann’s existence theorem relates to the circle method
Great video explained very well ❤
This math might be above my level, but it makes me hopeful. I was just wondering how to approach classifying ranges of relations as Rational or Irrational.
Very cool! Hope to see more facts from this profound theory.
Without algebra please
@@brendawilliams8062 did you get the idea of bridge from the video?
@@DanielRublev it’s harder with algebra
@@brendawilliams8062 maybe
Great job, but can you do it even simpler? like without using the jargon at all.
can you please do some concepts in representation theory , lie groups and that sort of math , great channel ❤❤
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
So is this like the Langlands program, just for Complex Analysis and Algebraic Geometry, as opposed to Number Theory and Geometry? Just trying to get my head around these different branches of Mathematics of which I clearly know so little!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
excelent video
"imaginary numbers" shouldve been called "orthogonal" numbers, then people could maybe understand how it's related to geometry
Love it.
Take it up with Descartes.😂
I call them "spinny numbers", because they are the best tool for the job of making 2D objects go spinny. (Naturally there are also 3D spinny numbers, which are rather famous, or more accurately infamous.)
And "complex numbers" could just be "2D numbers"
Yeah, I've never understood the "mystery" of imaginary numbers. It's just a mental construct that lets us model periodicity in a precise manner.
This is a great video for going to sleep. HIghly recommend to any insomniac.
5:33
Why do we need to explicitly evoke f(1/z)? Will lim z->inf f(z) not work?
Also, to check if a function is meromorphic at inf, is there no other way than to see this other than checking singularity of f(1/z)?
The notion of taking a limit as a value approaches infinity isn't well defined in the complex plane the same way it is for the real line.
On the real line, there's only really one way we can make a variable approach infinity (by making the variable bigger and bigger).
In the complex plane, variables can grow infinitely along an uncountably infinite amount of paths that move in different directions. We need to make a statement about what happens as z grows infinitely large in any of the possible directions. We're interested in what happens as |z| approaches infinity along any possible path.
Working with lim |z| -> infinity is technically sufficient to formulate the definition of a function being continuous, holomorphic, or meromorphic at infinity, but it's tricky to reason about a variable growing larger across the entire plane. We use the fact that as |z| approaches infinity, |1/z| approaches 0 to make the behavior we're interested in easier to reason about. By looking at the behavior of f(1/z) when |z| is small, we can study the behavior of f(z) as |z| approaches infinity by reasoning about the behavior of a function on a small disk, which is much more manageable than thinking about f's behavior as z grows larger in any of the possible directions.
@@chobes1827 thank you for your reply. That makes sense.
Exponentials are dual to logarithms.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
@@hyperduality2838 How high are you?
@@heeraksharma1224 Points are dual to lines -- the principle of duality in geometry.
If Riemann geometry is dual then this means that singularities (points) are dual.
Black holes = positive curvature singularities.
White holes (the big bang) = negative curvature singularities.
The definition of Gaussian negative curvature requires two dual points:-
en.wikipedia.org/wiki/Gaussian_curvature
The big bang is an infinite negative curvature singularity -- a Janus point/hole.
Two faces = duality.
The physicist Julian Barbour has written a book about Janus points/holes.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
Dark energy is repulsive gravity, negative curvature or hyperbolic space (a pringle) -- inflation.
The big bang an explosion is repulsive by definition -- negative curvature.
The point duality theorem is dual to the line duality theorem -- universal hyperbolic geometry.
The bad news is that Einstein threw his negative curvature solutions in the proverbial waste paper bin of history!
i need to do a repair on a jacket pocket but the best way to patch it would be with a riemann surface... sadly i can't find one in craft stores
i neeeed the next video!!
Great video as usual! Minor correction, at 11:59, you made a typo in "Riemann" (Reimann).
I think the way you glued the ends of the cylinder together at 7:20 will get you a Klein bottle
PS great video ❤
So it’s a neat twist on a binary system ? Just 1,0
Love the video!
thanks davide!
Finally he's back....
Interesting choice to talk about the complex torus and the p function and y^2 = x^3-x and NOT mention the term Elliptic curve 😉 Guess you didn't want to overload the video with even more topics
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
Nice!
Up next: the GAGA theorem
So my complex analysis exam is algebraic geometry in a trenchcoat?
Wow. More!
This the final conclusion of this reminds me a little of Umbral Calculus compared to polynomials...
Just checking but like. The arrow in that last image doesn’t go both ways, does it? Sure, every Riemann surface is an algebraic surface and that’s cool, but like. There are three-dimensional algebraic surfaces, but there are no three-dimensional Riemann surfaces, right? So there are some varieties that are not Riemann-able.
Is the final statement of this video false? Shouldn’t it be “SOME OF Complex Analysis SOME OF Algebraic Geometry”? Or do I need to watch the video again?
Example? Or is it that only if you make the right comparison or equality?
Ah yes, another glimpse of a mathematical world that is far too complex for my little mind.
At 11:55, Rain Man, oups Reimann, made his way in.
So every compact Riemann surface is an algebraic curve but is the other way true that every algebraic curve can be realized as a compact Riemann surface? If not these fields aren’t the same, just that these surfaces can be viewed equivalently in both but not all algebraic curves can be studied using complex analysis and not everything in complex analysis is a compact Riemann surface that can be studied in algebraic geometry. Therefore, I’m not sure I understand what the video is trying to conclude about them being the same and I’m just trying to understand that last point.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
9:10 elliptic curve?
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
yes that's why they are called elliptic curves, because ℘ and ℘' are elliptic functions
I'm just here for the colorful squares
beautiful
From here to Taniyama-Shimura!
Thanks!
Please do Bernhard the honor of spelling his last name correctly.
It's *Riemann*
(@ 12:00)
Riemann geometry is dual.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
@@hyperduality2838 Always so to the point.🤔🤣
@@oscargr_ Points are dual to lines -- the principle of duality in geometry.
If Riemann geometry is dual then this means that singularities (points) are dual.
Black holes = positive curvature singularities.
White holes (the big bang) = negative curvature singularities.
The definition of Gaussian negative curvature requires two dual points:-
en.wikipedia.org/wiki/Gaussian_curvature
The big bang is an infinite negative curvature singularity -- a Janus point/hole.
Two faces = duality.
The physicist Julian Barbour has written a book about Janus points/holes.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
Dark energy is repulsive gravity, negative curvature or hyperbolic space (a pringle) -- inflation.
The big bang an explosion is repulsive by definition -- negative curvature.
The point duality theorem is dual to the line duality theorem -- universal hyperbolic geometry.
The bad news is that Einstein threw his negative curvature solutions in the proverbial waste paper bin of history!
Oooh, are we going to do Ricci flow at some point? Or only Langlands stuff with fancy graphs like those meromorphics?
New Sub
greate thank you so much for share
wow!
Him: "f of zee equals exp of zed squared."
You must be a Bramerican.
Exponentials are dual to logarithms.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Real is dual to imaginary -- complex numbers are dual.
Injective is dual to surjective synthesizes bijective or isomorphism -- group theory.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Lie groups are dual to Lie algebras.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry or curvature is dual -- upper indices are dual to lower indices.
Positive curvature (convergence, syntropy) is dual to negative curvature (divergence, entropy) -- Gauss, Riemann geometry.
Subgroups are dual to subfields -- the Galois correspondence.
Elliptic curves are dual to modular forms.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- Category theory.
"Always two there are" -- Yoda.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Duality creates reality!
I think it might be good if you could get a pop filter. It sounds like I'm hearing a few too many pops
Plz make a video that.... You're counting 0 to you...(aleph null...)
FUCKING TEASE AT THE END
Very provocative indeed